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Introduction to Connectionist Approach in Artificial Intelligence

Learn about the connectionist approach in AI, including neural networks, perceptrons, backpropagation, and associative memory. Compare symbolic and sub-symbolic AI. Explore the brain metaphor and its influence on connectionist models.

richardmunoz
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Introduction to Connectionist Approach in Artificial Intelligence

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  1. CSC 550: Introduction to Artificial Intelligence Fall 2008 • Connectionist approach to AI • neural networks, neuron model • perceptrons threshold logic, perceptron training, convergence theorem single layer vs. multi-layer • backpropagation stepwise vs. continuous activation function • associative memory • Hopfield networks • parallel relaxation, relaxation as search

  2. Symbolic vs. sub-symbolic AI • recall: Good Old-Fashioned AI is inherently symbolic Physical Symbol System Hypothesis: A necessary and sufficient condition for intelligence is the representation and manipulation of symbols. • alternatives to symbolic AI • connectionist models – based on a brain metaphor model individual neurons and their connections properties: parallel, distributed, sub-symbolic examples: neural nets, associative memories • emergent models – based on an evolution metaphor potential solutions compete and evolve properties: massively parallel, complex behavior evolves out of simple behavior examples: genetic algorithms, cellular automata, artificial life

  3. Connectionist models (neural nets) • humans lack the speed & memory of computers • yet humans are capable of complex reasoning/action  maybe our brain architecture is well-suited for certain tasks • general brain architecture: • many (relatively) slow neurons, interconnected • dendrites serve as input devices (receive electrical impulses from other neurons) • cell body "sums" inputs from the dendrites (possibly inhibiting or exciting) • if sum exceeds some threshold, the neuron fires an output impulse along axon

  4. Brain metaphor • connectionist models are based on the brain metaphor • large number of simple, neuron-like processing elements • large number of weighted connections between neurons note: the weights encode information, not symbols! • parallel, distributed control • emphasis on learning • brief history of neural nets 1940's theoretical birth of neural networks McCulloch & Pitts (1943), Hebb (1949) 1950's & 1960's optimistic development using computer models Minsky (50's), Rosenblatt (60's) 1970's DEAD Minsky & Papert showed serious limitations 1980's & 1990's REBIRTH – new models, new techniques Backpropagation, Hopfield nets

  5. Artificial neurons • McCulloch & Pitts (1943) described an artificial neuron • inputs are either electrical impulse (1) or not (0) • (note: original version used +1 for excitatory and –1 for inhibitory signals) • each input has a weight associated with it • the activation function multiplies each input value by its weight • if the sum of the weighted inputs >= , • then the neuron fires (returns 1), else doesn't fire (returns 0) if wixi >= , output = 1 if wixi < , output = 0

  6. Computation via activation function • can view an artificial neuron as a computational element • accepts or classifies an input if the output fires • INPUT: x1 = 1, x2 = 1 • .75*1 + .75*1 = 1.5 >= 1  OUTPUT: 1 • INPUT: x1 = 1, x2 = 0 • .75*1 + .75*0 = .75 < 1  OUTPUT: 0 • INPUT: x1 = 0, x2 = 1 • .75*0 + .75*1 = .75 < 1  OUTPUT: 0 • INPUT: x1 = 0, x2 = 0 • .75*0 + .75*0 = 0 < 1  OUTPUT: 0 this neuron computes the AND function

  7. In-class exercise • specify weights and thresholds to compute OR • INPUT: x1 = 1, x2 = 1 • w1*1 + w2*1 >=  OUTPUT: 1 • INPUT: x1 = 1, x2 = 0 • w1*1 + w2*0 >=  OUTPUT: 1 • INPUT: x1 = 0, x2 = 1 • w1*0 + w2*1 >=  OUTPUT: 1 • INPUT: x1 = 0, x2 = 0 • w1*0 + w2*0 <  OUTPUT: 0

  8. Another exercise? • specify weights and thresholds to compute XOR • INPUT: x1 = 1, x2 = 1 • w1*1 + w2*1 >=  OUTPUT: 0 • INPUT: x1 = 1, x2 = 0 • w1*1 + w2*0 >=  OUTPUT: 1 • INPUT: x1 = 0, x2 = 1 • w1*0 + w2*1 >=  OUTPUT: 1 • INPUT: x1 = 0, x2 = 0 • w1*0 + w2*0 <  OUTPUT: 0 we'll come back to this later…

  9. Normalizing thresholds • to make life more uniform, can normalize the threshold to 0 • simply add an additional input x0 = 1, w0 = - • advantage: threshold = 0 for all neurons wixi >=   -*1 +wixi >= 0

  10. INPUT: x1 = 1, x2 = 1 • 1*-1 + .75*1 + .75*1 = .5 >= 0  OUTPUT: 1 • INPUT: x1 = 1, x2 = 0 • 1*-1 +.75*1 + .75*0 = -.25 < 1  OUTPUT: 0 • INPUT: x1 = 0, x2 = 1 • 1*-1 +.75*0 + .75*1 = -.25 < 1  OUTPUT: 0 • INPUT: x1 = 0, x2 = 0 • 1*-1 +.75*0 + .75*0 = -1 < 1  OUTPUT: 0 AND • INPUT: x1 = 1, x2 = 1 • 1*-.5 + .75*1 + .75*1 = 1 >= 0  OUTPUT: 1 • INPUT: x1 = 1, x2 = 0 • 1*-.5 +.75*1 + .75*0 = .25 > 1  OUTPUT: 1 • INPUT: x1 = 0, x2 = 1 • 1*-.5 +.75*0 + .75*1 = .25 < 1  OUTPUT: 1 • INPUT: x1 = 0, x2 = 0 • 1*-.5 +.75*0 + .75*0 = -.5 < 1  OUTPUT: 0 OR Normalized examples

  11. Perceptrons • Rosenblatt (1958) devised a learning algorithm for artificial neurons • start with a training set (example inputs & corresponding desired outputs) • train the network to recognize the examples in the training set (by adjusting the weights on the connections) • once trained, the network can be applied to new examples • Perceptron learning algorithm: • Set the weights on the connections with random values. • Iterate through the training set, comparing the output of the network with the desired output for each example. • If all the examples were handled correctly, then DONE. • Otherwise, update the weights for each incorrect example: • if should have fired on x1, …,xn but didn't, wi+= xi (0 <= i <= n) • if shouldn't have fired on x1, …,xn but did, wi-= xi (0 <= i <= n) • GO TO 2

  12. Example: perceptron learning • Suppose we want to train a perceptron to compute AND training set: x1 = 1, x2 = 1  1 x1 = 1, x2 = 0  0 x1 = 0, x2 = 1  0 x1 = 0, x2 = 0  0 randomly, let:w0 = -0.9, w1 = 0.6, w2 = 0.2 using these weights: x1 = 1, x2 = 1: -0.9*1 + 0.6*1 + 0.2*1 = -0.1  0 WRONG x1 = 1, x2 = 0: -0.9*1 + 0.6*1 + 0.2*0 = -0.3  0 OK x1 = 0, x2 = 1: -0.9*1 + 0.6*0 + 0.2*1 = -0.7  0 OK x1 = 0, x2 = 0: -0.9*1 + 0.6*0 + 0.2*0 = -0.9  0 OK new weights:w0 = -0.9 + 1 = 0.1 w1 = 0.6 + 1 = 1.6 w2 = 0.2 + 1 = 1.2

  13. Example: perceptron learning (cont.) using these updated weights: x1 = 1, x2 = 1: 0.1*1 + 1.6*1 + 1.2*1 = 2.9  1 OK x1 = 1, x2 = 0: 0.1*1 + 1.6*1 + 1.2*0 = 1.7  1 WRONG x1 = 0, x2 = 1: 0.1*1 + 1.6*0 + 1.2*1 = 1.3  1 WRONG x1 = 0, x2 = 0: 0.1*1 + 1.6*0 + 1.2*0 = 0.1  1 WRONG new weights: w0 = 0.1 - 1 - 1 - 1 = -2.9 w1 = 1.6 - 1 - 0 - 0 = 0.6 w2 = 1.2 - 0 - 1 - 0 = 0.2 using these updated weights: x1 = 1, x2 = 1: -2.9*1 + 0.6*1 + 0.2*1 = -2.1  0 WRONG x1 = 1, x2 = 0: -2.9*1 + 0.6*1 + 0.2*0 = -2.3  0 OK x1 = 0, x2 = 1: -2.9*1 + 0.6*0 + 0.2*1 = -2.7  0 OK x1 = 0, x2 = 0: -2.9*1 + 0.6*0 + 0.2*0 = -2.9  0 OK new weights: w0 = -2.9 + 1 = -1.9 w1 = 0.6 + 1 = 1.6 w2 = 0.2 + 1 = 1.2

  14. Example: perceptron learning (cont.) using these updated weights: x1 = 1, x2 = 1: -1.9*1 + 1.6*1 + 1.2*1 = 0.9  1 OK x1 = 1, x2 = 0: -1.9*1 + 1.6*1 + 1.2*0 = -0.3  0 OK x1 = 0, x2 = 1: -1.9*1 + 1.6*0 + 1.2*1 = -0.7  0 OK x1 = 0, x2 = 0: -1.9*1 + 1.6*0 + 1.2*0 = -1.9  0 OK DONE! EXERCISE: train a perceptron to compute OR

  15. Convergence • key reason for interest in perceptrons: Perceptron Convergence Theorem • The perceptron learning algorithm will always find weights to classify the inputs if such a set of weights exists. Minsky & Papert showed weights exist if and only if the problem is linearly separable intuition: consider the case with 2 inputs, x1 and x2 if you can draw a line and separate the accepting & non-accepting examples, then linearly separable the intuition generalizes: for n inputs, must be able to separate with an (n-1)-dimensional plane. • see http://www.avaye.com/index.php/neuralnets/simulators/freeware/perceptron

  16. Linearly separable firing depends on w0 + w1x1 + w2x2 >= 0 border case is when w0 + w1x1 + w2x2 = 0 i.e., x2 = (-w1/w2) x1 + (-w0 /w2) the equation of a line the training algorithm simply shifts the line around (by changing the weight) until the classes are separated why does this make sense?

  17. however, can compute XOR by introducing a new, hidden unit Inadequacy of perceptrons • inadequacy of perceptrons is due to the fact that many simple problems are not linearly separable

  18. Hidden units • the addition of hidden units allows the network to develop complex feature detectors (i.e., internal representations) • e.g., Optical Character Recognition (OCR) • perhaps one hidden unit • "looks for" a horizontal bar • another hidden unit • "looks for" a diagonal • another looks for the vertical base • the combination of specific • hidden units indicates a 7

  19. Building multi-layer nets • smaller example: can combine perceptrons to perform more complex computations (or classifications) • 3-layer neural net • 2 input nodes • 1 hidden node • 2 output nodes • RESULT? HINT: left output node is AND right output node is XOR HALF ADDER

  20. Hidden units & learning • every classification problem has a perceptron solution if enough hidden layers are used • i.e., multi-layer networks can compute anything • (recall: can simulate AND, OR, NOT gates) • expressiveness is not the problem – learning is! • it is not known how to systematically find solutions • the Perceptron Learning Algorithm can't adjust weights between levels • Minsky & Papert's results about the "inadequacy" of perceptrons pretty much killed neural net research in the 1970's • rebirth in the 1980's due to several developments • faster, more parallel computers • new learning algorithms e.g., backpropagation • new architectures e.g., Hopfield nets

  21. Backpropagation nets • backpropagation nets are multi-layer networks • normalize inputs between 0 (inhibit) and 1 (excite) • utilize a continuous activation function • perceptrons utilize a stepwise activation function output = 1 if sum >= 0 0 if sum < 0 • backpropagation nets utilize a continuous activation function output = 1/(1 + e-sum)

  22. Backpropagation example (XOR) • x1 = 1, x2 = 1 • sum(H1) = -2.2 + 5.7 + 5.7 = 9.2, output(H1) = 0.99 • sum(H2) = -4.8 + 3.2 + 3.2 = 1.6, output(H2) = 0.83 • sum = -2.8 + (0.99*6.4) + (0.83*-7) = -2.28, output = 0.09 • x1 = 1, x2 = 0 • sum(H1) = -2.2 + 5.7 + 0 = 3.5, output(H1) = 0.97 • sum(H2) = -4.8 + 3.2 + 0 = -1.6, output(H2) = 0.17 • sum = -2.8 + (0.97*6.4) + (0.17*-7) = 2.22, output = 0.90 • x1 = 0, x2 = 1 • sum(H1) = -2.2 + 0 + 5.7 = 3.5, output(H1) = 0.97 • sum(H2) = -4.8 + 0 + 3.2 = -1.6, output(H2) = 0.17 • sum = -2.8 + (0.97*6.4) + (0.17*-7) = 2.22, output = 0.90 • x1 = 0, x2 = 0 • sum(H1) = -2.2 + 0 + 0 = -2.2, output(H1) = 0.10 • sum(H2) = -4.8 + 0 + 0 = -4.8, output(H2) = 0.01 • sum = -2.8 + (0.10*6.4) + (0.01*-7) = -2.23, output = 0.10

  23. Backpropagation learning • there exists a systematic method for adjusting weights, but no global convergence theorem (as was the case for perceptrons) • backpropagation (backward propagation of error) – vaguely stated • select arbitrary weights • pick the first test case • make a forward pass, from inputs to output • compute an error estimate and make a backward pass, adjusting weights to reduce the error • repeat for the next test case testing & propagating for all training cases is known as an epoch • despite the lack of a convergence theorem, backpropagation works well in practice • however, many epochs may be required for convergence

  24. Backpropagation example • consider the following political poll, taken by six potential voters • each ranked various topics as to their importance, scale of 0 to 10 • voters 1-3 identified themselves as Democrats, voters 4-6 as Republicans based on survey responses, can we train a neural net to recognize Republicans and Democrats?

  25. Backpropagation example (cont.) • utilize the neural net (backpropagation) simulator at: • http://www.cs.ubc.ca/labs/lci/CIspace/Version4/neural/ • note: inputs to network can be real values between –1.0 and 1.0 • in this example, can use fractions to indicate the range of survey responses • e.g., response of 8  input value of 0.8 • APPLET IS FLAKEY - BE CAREFUL AND SPECIFY ALL INPUT/OUTPUT VALUES • make sure you recognize the training set accurately. • how many training cycles are needed? • how many hidden nodes?

  26. Backpropagation example (cont.) • using the neural net, try to classify the following new respondents

  27. Problems/challenges in neural nets research • learning problem • can the network be trained to solve a given problem? • if not linearly separable, no guarantee (but backpropagation is effective in practice) • architecture problem • are there useful architectures for solving a given problem? • most applications use a 3-layer (input, hidden, output), fully-connected net • scaling problem • how can training time be minimized? • difficult/complex problems may require thousands of epochs • generalization problem* • how know if the trained network will behave "reasonably" on new inputs? • backpropogation net trained to identify tanks in photos • trained on both positive and negative examples, very effective • when tested on new photos, failed miserably • WHY?

  28. Generalization problem • there is always a danger that the network will focus on specific features as opposed to general patterns (especially if many hidden nodes ? ) • to avoid networks that are too specific, cross-validation is often used • split training set into training & validation data • after each epoch, test the net on the validation data • continue until performance on the validation data diminishes (e.g., hillclimb) • suppose a network is trained to recognize digits: • training set for 1: • training set for 2: 1 1 1 1 2 2 2 2 • when the network is asked to identify: it comes back with 1. WHY? 2

  29. Neural net applications • pattern classification • 9 of top 10 US credit card companies use Falcon • uses neural nets to model customer behavior, identify fraud • claims improvement in fraud detection of 30-70% • Sharp, Mitsubishi, … -- Optical Character Recognition (OCR) • (see http://www.sund.de/netze/applets/BPN/bpn2/ochre.html ) • prediction & financial analysis • Merrill Lynch, Citibank, … -- financial forecasting, investing • Spiegel – marketing analysis, targeted catalog sales • control & optimization • Texaco – process control of an oil refinery • Intel – computer chip manufacturing quality control • AT&T – echo & noise control in phone lines (filters and compensates) • Ford engines utilize neural net chip to diagnose misfirings, reduce emissions • ALVINN project at CMU trained a neural net to drive • backpropagation network: video input, 9 hidden units, 45 outputs

  30. Interesting variation: Hopfield nets • in addition to uses as acceptor/classifier, neural nets can be used as associative memory – Hopfield (1982) • can store multiple patterns in the network, retrieve • interesting features • distributed representation • info is stored as a pattern of activations/weights • multiple info is imprinted on the same network • content-addressable memory • store patterns in a network by adjusting weights • to retrieve a pattern, specify a portion (will find a near match) • distributed, asynchronous control • individual processing elements behave independently • fault tolerance • a few processors can fail, and the network will still work

  31. Hopfield net examples • processing units are in one of two states: active or inactive • units are connected with weighted, symmetric connections positive weight  excitatory relation negative weight  inhibitory relation • to imprint a pattern • adjust the weights appropriately (no general algorithm is known, basically ad. hoc) • to retrieve a pattern: • specify a partial pattern in the net • perform parallel relaxation to achieve a steady state representing a near match

  32. Parallel relaxation • parallel relaxation algorithm: • pick a random unit • sum the weights on connections to active neighbors • if the sum is positive  make the unit active if the sum is negative  make the unit inactive • repeat until a stable state is achieved • this Hopfield net has 4 stable states • what are they? • parallel relaxation will start with an initial state and converge to one of these stable states

  33. Why does it converge? • parallel relaxation is guaranteed to converge on a stable state in a finite number of steps (i.e., node state flips) • WHY? Define H(net) =  (weights connecting active nodes) • Theorem: Every step in parallel relaxation increases H(net). • If step involves making a node active, this is because the sum of weights to active neighbors > 0. Therefore, making this node active increases H(net). • If step involves making a node inactive, this is because the sum of the weights to active neighbors < 0. Therefore, making this node active increases H(net). Since H(net) is bounded, relaxation must eventually stop  stable state

  34. Hopfield nets in Scheme • need to store the Hopfield network in a Scheme structure • could be unstructured, graph = collection of edges • could structure to make access easier (define HOPFIELD-NET '((A (B -1) (C 1) (D -1)) (B (A -1) (D 3)) (C (A 1) (D -1) (E 2) (F 1)) (D (A -1) (B 3) (C -1) (F -2) (G 3)) (E (C 2) (F 1)) (F (C 1) (D -2) (E 1) (G -1)) (G (D 3) (F -1))))

  35. Parallel relaxation in Scheme • (define (relax active) • (define (neighbor-sum neighbors active) • (cond ((null? neighbors) 0) • ((member (caar neighbors) active) • (+ (cadar neighbors) (neighbor-sum (cdr neighbors) active))) • (else (neighbor-sum (cdr neighbors) active)))) • (define (get-unstables net active) • (cond ((null? net) '()) • ((and (member (caar net) active) (< (neighbor-sum (cdar net) active) 0)) • (cons (caar net) (get-unstables (cdr net) active))) • ((and (not (member (caar net) active)) • (> (neighbor-sum (cdar net) active) 0)) • (cons (caar net) (get-unstables (cdr net) active))) • (else (get-unstables (cdr net) active)))) • (let ((unstables (get-unstables HOPFIELD-NET active))) • (if (null? unstables) • active • (let ((selected (list-ref unstables (random (length unstables))))) • (if (member selected active) • (relax (remove selected active)) • (relax (cons selected active)))))))

  36. > (relax '(a b)) (g d b) > (relax '(a b c e f)) (a c e f) > (relax '(a b c d e f g)) (b c d e g) > (relax '(a b c d)) (e g b c d) > (relax '(d c b a)) (g d b) • if you input a partial pattern, parallel relaxation will converge on a stored pattern • what can you say about the stored pattern that is reached? • is it in some sense the "closest" match? Relaxation examples • > (relax '()) • () • > (relax '(b d g)) • (b d g) • > (relax '(a c e f)) • (a c e f) • > (relax '(b c d e g)) • (b c d e g) parallel relaxation will identify stored patterns (since stable)

  37. Associative memory • a Hopfield net is associative memory • patterns are stored in the network via weights • if presented with a stored pattern, relaxation will verify its presence in the net • if presented with a new pattern, relaxation will find a match in the net • if unstable nodes are selected at random, can't make any claims of closeness • ideally, we would like to find the "closest" or "best" match fewest differences in active nodes? fewest flips between states?

  38. Parallel relaxation as search • can view the parallel relaxation algorithm as search • state is a list of active nodes • moves are obtained by flipping an unstable neighbor state

  39. Parallel relaxation using BFS • could use breadth first search (BFS) to find the pattern that is the fewest number of flips away from input pattern (define (relax active) (car (bfs-nocycles active))) (define (GET-MOVES active) (define (get-moves-help unstables) (cond ((null? unstables) '()) ((member (car unstables) active) (cons (remove (car unstables) active) (get-moves-help (cdr unstables)))) (else (cons (cons (car unstables) active) (get-moves-help (cdr unstables)))))) (get-moves-help (get-unstables HOPFIELD-NET active))) (define (GOAL? active) (null? (get-unstables HOPFIELD-NET active)))

  40. > (relax '(a b)) (g d b) > (relax '(a b c e f)) (a c e f) > (relax '(a b c d e f g)) (b c d e g) > (relax '(a b c d)) (g b d) > (relax '(d c b a)) (g d b) if you input a partial pattern, parallel relaxation will converge on "closest" pattern Relaxation examples • > (relax '()) • () • > (relax '(b d g)) • (b d g) • > (relax '(a c e f)) • (a c e f) • > (relax '(b c d e g)) • (b c d e g) parallel relaxation will identify stored patterns (since stable)

  41. Another example • consider the following Hopfield network • specify weights that would store the following patterns: AD, BE, ACE

  42. Additional readings • Neural Network from Wikipedia • NN applications from Stanford • Applications of adaptive systems from Peltarion • MSN Search's Ranking Algorithm uses a Neural Net by Richard Drawhorn • Recognition of face profiles from the MUGSHOT database using a hybrid connectionist/hmm approach by Wallhoff, Muller, and Rigoll

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